Space projects such as ALADIN demand a long-term stable operation in the space environment^[1], which is limited by what is commonly called the “fatigue” effect^[2,3] of the various optical components used in space lasers. Due to the fatigue effect, the damage threshold of these components decreases when the number of pulses increases. It is of great importance to identify the precursors and understand the damage mechanism^[4]. Wagner et al. described a model and discriminated two possible cases of fatigue: statistical pseudo-fatigue and cumulative material modifications^[5]. It is widely realized that the damage done to the transparent material in the nanosecond regime is caused by nanoscale absorptive defects^[6], which are hard to discover before the appearance of the damage. Statistical models applied to the analysis of laser damage probabilities are very useful and have been developed and improved for almost 40 years^[7–9]. Recently, Gouldieff et al. investigated the fatigue effects in bulk synthetic-fused silica with various wavelengths and laser spots^[10]. They found that the lifetime and fatigue dynamics were independent of the beam sizes used, but differed for two wavelengths. These models are also developed and corrected for multilayer coatings by considering the effect of the electric field (e-field) distribution^[11]. High-reflective (HR) coatings, which are an important transmission mirror in a high-power laser system, are constituted by alternating materials with a high and low index. The damage threshold and mechanism of the HR coatings under single-pulse or multiple-pulsed lasers have been investigated in our previous work^[12–14]. However, the defect analysis is initially based on the damage probability curve without considering the effect of the e-field. In addition, few studies exist that use the improved statistical model that corrects the effect of the e-field to analyze the defect in the damage experiments. In this paper, the fatigue effect of HR coatings constituted by hafnia (HfO2) and silica (SiO2) at 1064 nm are investigated.

The fundamental idea of the statistical model is based on the assumption that the damage probability P(F) is equivalent to the probability of a defect appearing inside the laser spot and obeying the Poisson Distribution^[7]. So, P(F)=1−exp[−N(F)],(1)where F and N(F) are the incident fluence in the free space and the expected number of defects under the fluence F, respectively.

Although the effect of the e-field is corrected by integrating the defect in the effective volume of the single layer and summing over all the layers^[11], further correction can be made with the defect distribution.

The real energy density in each high and low unit can be deduced with the incident fluence F and |E(z)|2. Allowing for the features of the HR coatings, we assume that the defects are concentrated in the bilayer unit of the coating materials with the same threshold distribution g(T). Thus, the expected defect number can be given by N(F)=g(T)V(F),(2)V(F)=∫0ZmaxSeffln[F|E(z)|2T]dz,(3)g(T)=2d0ΔT02πexp[−12(T−T0ΔT0/2)2],(4)where z, Seff, and |E(z)|2 are the depth relative to the air-film interface, the effective laser spot area, and the normalized squared e-field intensity at the ith unit, respectively. The major defect parameters in g(T) are the average damage threshold T0, the threshold standard deviation ΔT, and the defect density d. By summing over all the units, the total expected defect number under the fluence F can be expressed as N(F)=∑iρiVi(F).(5)

By combining Eqs. (1), (2), and (5), the damage probability curve for the HR coatings can be calculated.

To investigate the fatigue effect, HR coatings constituted by HfO2 and SiO2 are deposited using the e-beam evaporation method. The stack formula of the HR coatings is substrate/(HL)12H4L/air with the center wavelength of 1064 nm, in which H and L denote the high and low index material, respectively. The refractive indices of the HfO2 and SiO2 used in our coatings are 1.92 and 1.46. The reflectance of the HR coatings at 1064 nm is more than 99.5% at the normal incidence. The damage test is implemented by a Nd:YAG laser operated at 1064 nm with a pulse width of 12 ns in single longitudinal mode with up to a 5 Hz repetition rate. The 1/e2 spot diameters on the x and y axis measured via the knife-edge method are 260 and 300 μm. The detail introduction of the damage test system can be found in our previous works^[13]. The damage test is completed with different shot numbers from 1 to 1000.

The normalized squared e-field intensity distribution of the HfO2/SiO2 coatings at 1064 nm is calculated and shown in Fig. 1. Due to the measurement of the laser fluence in the air, the factual fluence in the coating material is rather unstable, which can be described as a typical standing-wave e-field. In each HL bilayer unit, the e-field changes rapidly from zero to the maximum and then decreases back to zero.

The damage results under different shot numbers are fitted with the statistical model mentioned above, as shown in Fig. 2(a). The shape of the damage changes when the shot numbers are less than 60. However, the damage probability curve tends to be saturated with higher shot numbers. The variations of the damage threshold of 0% (red line) and 100% (black line) probability with the shot numbers are depicted in Fig. 2(b). As seen from the Fig. 2(b), the damage threshold of the 0% probability decreases from 50 to 23J/cm2, which is the typical fatigue effect of devices under multiple-pulse laser damage. The fatigue effect is further investigated with the analysis of the relationship between the damage threshold of 100% probability and the shot numbers. From Fig. 2(b), it can be found that the black line decreases much faster than the red line, which means that the influence of the shot numbers is mainly on the 100% damage threshold. Therefore, we can conclude that the multiple irradiations on our HR coatings mainly affected the distribution of the high threshold precursors.

According to the experimental data and the fitted model, the average damage threshold T0, the threshold standard deviation ΔT, and the defect density d can be extracted. The results of T0 versus the shot number are presented in Fig. 3(a). It can be seen that T0 drops rapidly as the shot numbers increase. When the shot numbers reach 60, the defects differentiate into two classes. One continues to deteriorate under subsequent irradiation, which can be taken as a proof of the “fatigue effect” mentioned previously. The other remains stable. However, the overall defect level is certain, and the competition between the two kinds of defects will gradually saturate the fatigue effects. The relationship between the threshold standard deviation ΔT and the shot numbers is shown in Fig. 3(b). Combined with the information on the threshold standard deviation, one can see that when the fatigue effects dominate, lower threshold defects will significantly decrease the fluctuation range of the damage threshold. This means that the deterioration of a part of the high threshold defect causes the overall threshold level to span to the lower level and become concentrated, which is also a reflection of the fatigue effect. When the deterioration process described above becomes saturated, all of the fatigue effects become saturated. The long lifetime of the device is decided by the distribution level of the saturated defect.

In addition, the relationship between the densities of all potential defects and number of shots is also shown in Fig. 4. The defect density here is described with the assumption that all of the potential defects are motivated under a very high irradiation fluence that is not equal to the fluence for 100% damage probability. It can be seen that under low shot numbers, the defect density initially increases, which reveals more and more defects under the maximum laser irradiation fluence. But with the saturation and steadying of the deterioration, the density of the defects gradually stabilizes.

In order to visually present the conversion process of the high threshold defect to the low threshold defect, the defect density distributions versus the intrinsic threshold with different shot numbers ranging from 1 to 1000 are shown in Fig. 5. Here, the defect density distribution g(T) extracted from the fitted model can reflect the intrinsic damage performance of the coating materials. The change in the high threshold defects in the curve is the most significant when the shot numbers increase from 1 to 60. It can be seen clearly that the process of the conversion of the high damage precursors tends to be saturated. The phenomenon described here is in keeping with the fatigue effects mentioned above. When the number of shots is more than 60, the defects distribution tends to the level of saturation, and with the increase in the pulse, the device’s longevity does not have any significant changes. According to the recent research, the different defect classes do not exist independently, and one might be converted to another^[10,15,16].

In conclusion, we use a simplified method to investigate the relationship between the fatigue effect and shot numbers for the HR coatings. The effect of the standing-wave e-field is corrected in the model. The defect density, average damage threshold, and threshold standard deviation are extracted from the damage probability fitted curve. Two different classes of defects are found. The differentiation of the high threshold precursors can well explain the decrease in the laser-induced damage threshold. It is concluded that the fatigue effect of the HR coatings under multiple pulses can be attributed to the conversion of the high damage threshold precursors. The saturation effect can also be observed when the number of shots in high enough.